设{an}是公比不等于1的等比数列,Sn是其前n项和,若a1,2a7,3a4成等差数列,求证12S3,S6,S12-S6成等比数列

题目

设{an}是公比不等于1的等比数列,Sn是其前n项和,若a1,2a7,3a4成等差数列,求证12S3,S6,S12-S6成等比数列

答案

q=公比≠1a7=a4q^34a7=a1+3a4=4a4q^3a4=a1/(4q^3-3)=a1q^3,Q=q^31=(4Q-3)Q=4Q^2-3Q4Q^2-3Q-1=0(4Q+1)(Q-1)=0Q=-1/4q=(-1/4)^(1/3)令z=1-q12S3=12a1(1-q^3)/(1-q)=15a1/(1-q)=15a1/zS6=a1(1-1/16)/(1-q)=15a1/16zS12-...